Understand the purpose and uses of propositional logic.
Understand the meaning, the symbols, and the uses of the four logical connectives -- conjunction, disjunction, negation, and conditional.
Know the definition of statement and the distinction between simple and compound statements.
Know how to translate simple statements into symbolic form.
Know how to construct a truth table and how to use it to test the validity of arguments.
Know the situations in which conjunctions, disjunctions, negations, and conditionals are truth or false.
Understand the structure of conditional statements and the various ways in which they can be expressed.
Checking for Validity
Know how to determine the validity of very simple arguments using truth tables.
Be able to use parentheses to express statements in symbolic form.
Be able to use the short method to evaluate complex arguments.
Chapter summary
In propositional logic we use symbols to stand for the relationships between statements -- that is, to indicate the form of an argument. These relationships are made possible by logical connectives such as conjunction (and), disjunction (or), negation (not), and conditional (If?then?). Connectives are used in compound statements, each of which is composed of at least two simple statements. A simple statement is a sentence that can be either true or false.
To indicate the possible truth values of statements and arguments, we can construct truth tables, a graphic way of displaying all the truth value possibilities. A conjunction is false if at least one of its statement components (conjuncts) is false. A disjunction is still true even if one of its component statements (disjuncts) is false. A negation is the denial of a statement. The negation of any statement changes the statement's truth value to its contradictory (false to true, and true to false). A conditional statement is false in only one situation -- when the antecedent is true and the consequent is false.
The use of truth tables to determine the validity of an argument is based on the fact that it's impossible for a valid argument to have true premises and a false conclusion. A basic truth table consists of two or more guide columns listing all the truth value possibilities, followed by a column for each premise and the conclusion. We can add other columns to help us determine the truth values of components of the argument.
Some arguments are complex when variables and connectives are combined into larger compounds and when the number of variables increases. To prevent confusion, we can use parentheses in symbolized arguments to show how statement or premise components go together.
You can check the validity of arguments not only with truth tables but also with the short method. In this procedure we try to discover if there is a way to make the conclusion false and the premises true by assigning various truth values to the argument's components.
Chapter 06
Chapter Objectives
Connectives and Truth Values
Checking for Validity
Chapter summary
In propositional logic we use symbols to stand for the relationships between statements -- that is, to indicate the form of an argument. These relationships are made possible by logical connectives such as conjunction (and), disjunction (or), negation (not), and conditional (If?then?). Connectives are used in compound statements, each of which is composed of at least two simple statements. A simple statement is a sentence that can be either true or false.
To indicate the possible truth values of statements and arguments, we can construct truth tables, a graphic way of displaying all the truth value possibilities. A conjunction is false if at least one of its statement components (conjuncts) is false. A disjunction is still true even if one of its component statements (disjuncts) is false. A negation is the denial of a statement. The negation of any statement changes the statement's truth value to its contradictory (false to true, and true to false). A conditional statement is false in only one situation -- when the antecedent is true and the consequent is false.
The use of truth tables to determine the validity of an argument is based on the fact that it's impossible for a valid argument to have true premises and a false conclusion. A basic truth table consists of two or more guide columns listing all the truth value possibilities, followed by a column for each premise and the conclusion. We can add other columns to help us determine the truth values of components of the argument.
Some arguments are complex when variables and connectives are combined into larger compounds and when the number of variables increases. To prevent confusion, we can use parentheses in symbolized arguments to show how statement or premise components go together.
You can check the validity of arguments not only with truth tables but also with the short method. In this procedure we try to discover if there is a way to make the conclusion false and the premises true by assigning various truth values to the argument's components.